Question

The maximum value of ${}^{2n}{C}_r$ is equal to

• A. ${}^{2n-1}{C}_r$
• B. $2\left({}^{2n-1}{C}_r\right)$
• C. ${}^{2n-1}{C}_{r-1}$
• D. $2\left({}^{2n}{C}_r\right)$

Answer 1

The correct option is B $2\left({}^{2n-1}{C}_r\right)$

${}^{2n}{C}_r$ will be maximum, when $r=\frac{2n}{2}=n$
Now, ${}^{2n}{C}_r={}^{2n}{C}_n=\frac{(2n!)}{(n!{)}^2}$
$=\frac{2n\cdot (2n-1)\cdots 2\cdot 1}{n!\left(\right(n\cdot (n-1)\cdots 2\cdot 1)}$
$=\frac{2\cdot (2n-1)!}{n!(n-1)!}$
$=2\left({}^{2n-1}{C}_n\right)$